Generalizations of Boolean products for lattice-ordered algebras P. Jipsen Chapman University, Department of Mathematics and Computer Science, Orange, CA 92866, USA

Dedicated to Franco Montagna on the occasion of his 60th birthday

Abstract It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw -algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras. Key words: Residuated lattices, generalized BL-algebras, basic logic, generalized MV-algebras, posets 2000 MSC: 06F05, 06D35, 03G10, 03G25,

1. Introduction Topological dualities have been very effective tools for various classes of algebras, such as Boolean algebras with Boolean spaces as duals, distributive lattices with Priestley spaces as duals, and Heyting algebras with Esakia spaces as duals. Boolean spaces have also been applied to the representation of algebras by Boolean powers and (weak) Boolean products, where the latter are also known as algebras of global sections of sheaves of algebras over Boolean spaces [2]. In Section 2 we recall the concept of (weak) Boolean product, and define the poset product for algebras of any signature with two constants 0,1 (previously the latter notion was defined only for residuated lattices [10]). We prove that under mild assumptions on the basic operations of the algebras, the poset product is a subalgebra of the direct product. Section 3 contains general results

Email address: [email protected] (P. Jipsen)

Preprint submitted to Elsevier

November 24, 2008

about direct decompositions of integral bounded unital `-groupoids, based on the Boolean center of complemented elements. In the next section we restate an embedding result, proved for integral GBL-algebras in [11], so that it applies to FLw -algebras in general. Theorem 12 shows that an FLw -algebra with any finite subalgebra of strongly central elements (i.e. elements c that satisfy c ∧ x = cx = xc for all x) decomposes as a poset product indexed by the dual poset of join irreducible elements of the subalgebra, which generalizes a similar result of [10] for finite GBL-algebras. Finally in Section 5 we combine Boolean products and poset products by defining the concept of Priestley product and Esakia product. The latter notion is used to show that any bounded n-potent GBL-algebra is an Esakia product of simple n-potent MV-algebras. 2. Boolean products and poset products Let {Ai : i ∈ X} be a family of algebras with the same fundamental opQ eration symbols from a set F. The direct (cartesian) product i∈X A of this i S family of algebras is of course the set of all functions f : X → i∈X Ai such that f (i) ∈ Ai for all i ∈ X (i.e. choice Q functions), with the operations defined pointwise, and with projections πj : i∈X Ai  Aj . It is not often the case that an algebra can be expressed as a direct product of simpler algebras, so various generalizations of products are used to obtain more widely applicable representation results. E.g. Birkhoff’s subdirect product represents algebras as subalgebras of direct products for which the projections are still surjective. Recall that a Boolean space is a set with a Boolean topology, defined as a topology that is compact and totally disconnected (i.e. distinct elements are separated by clopen sets, hence every Boolean space is Hausdorff). By Stone duality, clopen sets of a Boolean space X form a Boolean algebra AX , and the set XA of ultrafilters of a Boolean algebra A carry a natural Boolean topology such that XAXQ∼ = X and AXA ∼ = A. A weak Boolean product is a subdirect product A ≤ i∈X Ai for which there exists a Boolean topology on the index set X such that for all f, g ∈ A (i) the equalizer [[f = g]] = {i ∈ X : f (i) = g(i)} is open and (ii) for all clopen U , f |U ∪ g|X−U ∈ A If “open” is replaced by “clopen” in (i) then A is a Boolean product of {Ai : i ∈ X}. The Boolean power of an algebra B over a Boolean space X = (X, τ ) is B[X]∗ = {f ∈ B X : f −1 [{b}] is open for all b ∈ B} i.e. the set of continuous functions from X to B, where B is considered to have the discrete topology. Every Boolean power is a Boolean product (see e.g. [2]), and if X is a finite set then both concepts reduce to the direct product (since any function on a finite domain can be constructed from a finite union of restrictions of functions in a subdirect product). Boolean products have been used in many settings to derive powerful decidability results and representation 2

results for classes of algebras, see e.g. [3], [2] for discriminator algebras, [5] for lattices, [4] for MV-algebras, [7] for BL-algebras. The poset product (introduced for residuated lattices in [10] as dual poset sum) uses a partial order on the index set to define a subset of the direct product. Specifically, let X = (X, ≤) be a poset, and assume the algebras Ai have two distinct constant operations denoted 0, 1. A labeling of X is a choice function S f : X → i∈X Ai . An antichain labeling f of X (or ac-labeling for short) is a labeling that satisfies f (i) = 0 or f (j) = 1

for all i < j in X.

The poset product of {Ai : i ∈ X} is Y Y Ai = {f ∈ Ai : f is an ac-labeling}. X

i∈X

The poset product is distinguished visually from the direct product since the index set is a poset X rather than just a set X. The terminology “antichain labeling” is explained by the following observation. Lemma 1. Let X be a poset, and {Ai : i ∈ X} a family of algebras with constants 0, 1. For a labeling f of X the following are equivalent. (i) f is an antichain labeling. (ii) {i ∈ X : f (i) ∈ / {0, 1}} is a (possibly empty) antichain of X, f −1 [{0}] is a downset of X and f −1 [{1}] is an upset of X. S Proof. (i)⇒(ii): Assume f : X → i∈X Ai is an ac-labeling, and consider i, j ∈ X. If f (i), f (j) ∈ / {0, 1} then they are incomparable, hence the set of all elements labeled neither 0 nor 1 is an antichain. If f (j) = 0 6= 1 and i < j then f (i) = 0 hence f −1 [{0}] is a downset, and dually for f −1 [{1}]. (ii)⇒(i): Assume (ii), suppose f is a labeling, and let i < j. If f (i) 6= 0 then i is in the antichain of elements labeled neither 0 nor 1, or f (i) = 1. In either case we must have f (j) = 1, hence f is an ac-labeling. For every labeling f of X there are two “projections” p0 (f ) and p1 (f ) into the poset product defined by ( f (i) if f (j) = 1 for all j > i p0 (f )(i) = 0 otherwise ( f (i) if f (j) = 0 for all j < i p1 (f )(i) = 1 otherwise Now each basic operation o ∈ F is defined on the poset product A by Q

oA (f1 , . . . , fn ) = p0 (o

3

i∈X

Ai

(f1 , . . . , fn ))

Q

where o i∈X Ai is the usual pointwise operation on the direct product. A poset power is a poset product where all the factor algebras are identical to an algebra Q B, in which case X Ai is denoted by BX . Note that a poset product is not, in general, a subalgebra of the direct product. However, with some mild assumptions on the basic operations of the algebras, the following result shows that the projections have no effect, and hence the poset sum is closed under pointwise defined operations. An element c in an algebra A is an idempotent of the operation o if oA (c, c, . . . , c) = c, and the operation is strict with respect to c if oA (x1 , . . . , xi−1 , c, xi+1 , . . . , xn ) = c for all i ∈ {1, . . . , n} and all x1 , . . . , xn ∈ A. Q Lemma 2. Let A = X Ai for some poset X and family {Ai : i ∈ X}. If 0, 1 are idempotents of o and if o is strict with respect to 0 in each Ai or strict with respect to 1 in each Ai then oA is computed pointwise in A. Proof. Suppose 0, 1 are distinct idempotents and o is strict with respect to 0 in each Ai . For f1 , ..., fn ∈ A, let f be the result of applying o to f1 , . . . , fn pointwise and consider i < j in X. If fk (i) = 0 for some k ∈ {1, . . . , n} then f (i) = 0 since o is strict, and if fk (i) 6= 0 for all k ∈ {1, . . . , n} then fk (j) = 1 for all k and hence f (j) = 1 since 1 is an idempotent. Therefore f is an aclabeling, and the proof for o strict with respect to 1 is similar. It follows that p0 (f ) = p1 (f ) = f ∈ A. Our main application of the poset product is to bounded lattice-ordered algebras, and specifically to bounded residuated lattices. In the most general setting, a lattice-ordered algebra (or `-algebra) is any universal algebra that has a lattice reduct. However, one often assumes that the operations preserve joins or meets, or interchange joins or meets, in each argument. For example, `groupoids, unital `-groupoids, `-monoids and `-groups are defined as groupoids, unital groupoids, monoids and groups that are expanded with lattice operations and satisfy the identities x(y ∨ z) = xy ∨ xz and (x ∨ y)z = xz ∨ yz. They are bounded if there are constants ⊥, > denoting the bottom and top element of the lattice reduct. A bounded residuated lattice A = (A, ∧, ∨, ·, \, /, 1, ⊥, >) is a lattice-ordered monoid (A, ∧, ∨, ·, 1) such that for all x, y, z ∈ A x·y ≤z

iff

x ≤ z/y

iff

y ≤ x\z

and ⊥, > are the bottom and top element of A (see e.g. [8]). For bounded residuated lattices the operations ∧, ∨, · satisfy the assumption of the previous lemma (with 0, 1 replaced by ⊥, >), while \, / do not. The next result implies that the poset product of a family of bounded residuated lattices is again a bounded residuated lattice, and this motivates our choice of p0 (rather than p1 ) in the definition of operations on poset products. Lemma 3. Let f be a labeling of a poset X and assume that the algebras Ai are partially ordered with 0 and Q 1 as bottom and top elements respectively. Then p0 (f ) is the largest element of X Ai that is pointwise less or equal to f , and likewise p1 (f ) is the smallest element that is pointwise greater or equal to f . 4

3. Direct decompositions and Boolean products of FLw -algebras Mostly we consider integral bounded unital `-groupoids (or ibu`-groupoids for short), i.e. they have the identity element 1 as top element, and in this case the bottom element is denoted by 0. A residuated `-groupoid (or r`-groupoid) is an `-groupoid for which the residuals \, / exist relative to the groupoid operation. A FLw -algebra is a residuated integral bounded `-monoid (see e.g. [8]). A subset F of a residuated lattice A is a filter if F is up-closed, 1 ∈ F and F is closed under the monoid operation and the meet operation. A filter F is normal if it is closed under conjugation, i.e. x ∈ F and y ∈ A

y\(xy), (yx)/y ∈ F . S For a ∈ A and S ⊆ A, we let ↓A a = {x ∈ A : x ≤ a}, ↓A S = {↓A a : a ∈ S}, and ↑A a, ↑A S are defined dually (A is often omitted). For any residuated lattice, the lattice of normal filters is isomorphic to the congruence lattice via θ 7→ ↑{x : (x, 1) ∈ θ} and F 7→ {(x, y) : x\y, y\x ∈ F }. The congruence class of an element x ∈ A with respect to the congruence induced by the filter F is denoted by x/F . A normal residuated lattice is one in which every filter is normal. For example every commutative residuated lattice is normal. Before characterizing poset decompositions we consider some results about direct decompositions. An element c in an ibu`-groupoid A is complemented if there exists c0 ∈ A such that c∧c0 = 0 and c∨c0 = 1. The Boolean center of A is the set B(A) of all complemented elements. The next result generalizes similar results for MV-algebras [4] and BL-algebras [7]. The first part is essentially from [1]. imply

Lemma 4. Let A be an ibu`-groupoid and let c ∈ B(A). Then (i) x ∧ c = xc = cx for all x ∈ A, hence the Boolean center is a Boolean sublattice of central idempotent elements. (ii) If A is a residuated ibu`-groupoid then B(A) is also closed under the residuals, the complement of c is −c = 0/c = c\0 and c\x = x/c = −c ∨ x for all c ∈ B(A) and x ∈ A. Proof. (i) Suppose A is an ibu`-groupoid and c∧d = 0, c∨d = 1. By integrality cx ≤ c ∧ x = (c ∨ d)(c ∧ x) = c(c ∧ x) ∨ d(c ∧ x) ≤ cx ∨ 0 = cx, and similarly xc ≤ x ∧ c ≤ xc. Suppose we also have a ∧ b = 0, a ∨ b = 1. To see that B(A) is a sublattice of A, it suffices to show that a ∨ c and b ∧ d are complements: (a ∨ c) ∧ (b ∧ d) = (a ∨ c)bd = abd ∨ cbd = 0 and (a ∨ c) ∨ (b ∧ d) = a ∨ c ∨ bd = a ∨ c ∨ bc ∨ bd = a ∨ c ∨ b(c ∨ d) = a ∨ c ∨ b = 1. Now B(A) is complemented by definition, and it is a distributive lattice since · distributes over ∨, hence it is a Boolean lattice. (ii) For complements c, d and any x ∈ A we have c\x = (c ∨ d)(c\x) = c(c\x) ∨ d(c\x) ≤ x ∨ d. On the other hand c(x ∨ d) = cx ∨ cd ≤ x implies x ∨ d ≤ c\x. Hence c\x = d ∨ x, and for x = 0 we obtain −c = c\0 = d. Therefore c\x = −c ∨ x for all x ∈ A. The results for / follow similarly. 5

For an ib(r)u`-groupoid A and an element c ∈ B(A), define the relativized subalgebra Ac with universe Ac = ↓c, unit 1Ac = c, operations ∧, ∨, · restricted from A, and a\b = (a\A b) ∧ c, a/b = (a/A b) ∧ c for all a, b ∈ ↓c. Lemma 5. For any ib(r)u`-groupoid A and any c ∈ B(A), the relativized subalgebra Ac is an ib(r)u`-groupoid. If A is an FLw -algebra then the map f : A → Ac given by f (a) = ac is a homomorphism, hence Ac satisfies all identities that hold in A. Proof. By (i) of the preceding lemma, Ac has c as a unit and is closed under ∧, ∨, ·, hence it is an ibu`-groupoid. If A has residuals then for all a, b, x ∈ Ac we have ax ≤ b

iff

x ≤A a\A b and x ≤A c

iff

x ≤ a\b,

and similarly a/b = (a/A b) ∧ c, whence \, / are residuals of · in Ac. Now f (1) = 1c = 1Ac , (a ∧ b)c = a ∧ b ∧ c = ac ∧ bc and (a ∨ b)c = ac ∨ bc hence f preserves ∧, ∨. If · is associative then (ab)c = abcc = (ac)(bc). In any residuated lattice x\y ≤ zx\zy, hence f (a\A b) ≤ f (a)\f (b). For the opposite inequality, we have ac(ac\A bc) ≤ bc ≤ b, so c(ac\A bc) ≤ a\A b, and therefore (ac\A bc) ∧ c ≤ (a\A b)c. This shows f (a)\f (b) ≤ f (a\A b). Theorem 6. If A is an FLw -algebra and if c, d ∈ B(A) are complements then A∼ = Ac × Ad. Proof. Consider the map h : A → Ac×Ad defined by h(a) = (a∧c, a∧d). The preceding two lemmas show that h is a homomorphism, and h has an inverse given by (x, y) 7→ x ∨ y since ac ∨ ad = a(c ∨ d) = a and for x ≤ c, y ≤ d we have ((x ∨ y)c, (x ∨ y)d) = (xc ∨ yc, xd ∨ yd) = (x, y). Conversely, any direct decomposition of an ib(r)u`-groupoid is obtained in this way, since the elements (0, 1), (1, 0) are complements. Corollary 7. An FLw -algebra is directly indecomposable iff its Boolean center contains only the elements {0, 1}. The preceding results about direct decompositions are useful for a characterization of (weak) Boolean products of FLw -algebras. We first recall a general characterization of weak Boolean products in terms of Boolean algebras of factor congruences from [13]. A (weak) Boolean decomposition of A is an isomorphism from A to a (weak) Boolean product. A pair θ, ψ of congruences of A are called factor congruences if θ ∩ ψ = idA and θ ◦ ψ = A2 . A Boolean algebra of factor congruences is a set of factor congruences that is a Boolean algebra, with ∩ and ◦ as lattice operations. Theorem 8. Let A be an algebra. (i) Suppose K is a Boolean algebra A. For each S of factor congruences on Q prime filter F of K, let θF = (K − F ) and define ε : A → F ∈XK A/θF by ε(a)(F ) = a/θF . Then ε is a weak Boolean decomposition of A. 6

Q (ii) If X is a Boolean space and ε0 : A → i∈X Ai is any weak Boolean decomposition then there exists a unique Boolean algebra K of factor congruences, a homeomorphism k : X → XK , and isomorphisms hi : A i ∼ = Q A/θk(i) such that hi πi ε0 = πk(i) ε, where ε : A → i∈X A/θk(i) s given by ε(a)(i) = a/θk(i) . The algebra K in (ii) is the set of congruences ψU = ∩{ker(πi ε0 ) : i ∈ U } where U ranges over the clopen sets of X. For an FLw -algebra A the algebra of all factor congruences is isomorphic to B(A). The following result generalizes Theorem 2.1 in [7]. Corollary 9. Let A be a weak Boolean product of a nonempty family {Ai : i ∈ X} of non-trivial FLw -algebras over a Boolean space X and let C = {f ∈ A : f [X] ⊆ {0, 1}}. Then (i) (ii) (iii) (iv)

C is a subalgebra of B(A), the map k(i) = {f ∈ C : f (i) = 1} is a homeomorphism from X onto XC , Ai is isomorphic to A/↑k(i), and C coincides with B(A) iff all algebras Ai are directly indecomposable.

Conversely, suppose A is a nontrivial FLw -algebra and C is a subalgebra of B(A). Then A is isomorphic to a weak Boolean product of {A/↑F : F ∈ XC }. Proof. (i) holds since f ∈ C implies f \0 is a complement of f , and (ii) follows from the observation that the algebra AX of clopen subsets of X is isomorphic to C∼ = AXC . The isomorphism in (iii) follows from (ii) of the preceding theorem, and the converse is from part (i) of the same result. 4. Embeddings and representations via poset products A generalized BL-algebra or (GBL-algebra for short) is a residuated lattice that is divisible, i.e. satisfies x≤y

⇒

x = (x/y)y = y(y\x).

This property is equivalent to an identity (replace x by x ∧ y), and implies that there are no idempotent elements above 1. Hence any bounded GBL-algebra is integral, and we again denote the bottom element by 0. As examples we list the following subvarieties: • BL-algebras are bounded GBL-algebras that satisfy commutativity (xy = yx) and prelinearity (x\y ∨ y\x = 1), • Heyting algebras are bounded GBL-algebras in which all elements are idempotent (whence xy = x ∧ y), • GMV-algebras are GBL-algebras that satisfy x ∧ y = x/(y\x) = (x/y)\x), • pseudo MV-algebras are bounded GMV-algebras, 7

• MV-algebras in addition satisfy commutativity xy = yx, and • Boolean algebras are the intersection of Heyting algebras and (pseudo)MV-algebras. We now recall a result from [11] that gives sufficient conditions for an algebra to be embeddable into a poset product. There it is proved for integral GBLalgebras, and the factors are assumed to be totally ordered GMV-algebras. Since they need not have a lower bound, the factors are first embedded into pseudo MV-algebras. Here we state the result for FLw -algebras in general, but note that the proof is essentially the same. The ordinalQ sum of two algebras B0 , B1 , each with constants 0, 1, is defined as B0 ⊕ B1 = 2∂ Bi , where 2∂ = {0, 1} is the two element poset with 1 < 0. For ib(r)ul-groupoids this agrees with the usual definition of (amalgamated) ordinal sum where all elements of B0 are less or equal to all elements of B1 . Theorem 10. Let A be a FLw -algebra, X a poset, and {Fi : i ∈ X} a family of normal filters of A such that for all i ∈ X A/Fi = Bi ⊕ Ci where Bi , Ci are FLw -algebras, c ⊆ Fj for all c ∈ Ci and all j > i, for all a ∈ / Fi there exists j ≥ i such that a/Fj ∈ Cj − {1/Fj }, T F = {1}. i i∈X Q Then A embeds into the poset product X Ci . (i) (ii) (iii) (iv)

In [11] this theorem is used to prove that every integral normal GBL-algebra embeds into a poset product of totally ordered integral bounded GMV-algebras. The key result that enables this application is the Blok-Ferreirim decomposition theorem for subdirectly irreducible integral normal GBL-algebras proved in [10]: every such algebra is isomorphic to an ordinal sum B ⊕ W where W is a nontrivial totally ordered integral GMV-algebra and B is an integral GBL-algebra. An algebra A is poset indecomposable if whenever A is isomorphic to a poset Q product X Ai there exists i ∈ X such that A ∼ = Ai . In [9] it is shown that every finite GBL-algebra is isomorphic to a (uniquely determined) poset product of totally ordered integral GMV-algebras, which are poset indecomposable. In the next section we augment poset products with a Boolean topology on the index poset, with the aim of extending the representation of finite GBL-algebras to a larger class of algebras. For a residuated lattice A we define the set of strongly central elements IA = {a ∈ A : a ∧ x = ax = xa for all x ∈ A}. Recall from [9] that if A is a GBL-algebra then IA is a subalgebra of A. For bounded GBL-algebras, IA is in fact a Heyting algebra, and B(A) is the subalgebra of complemented elements of IA . For MV-algebras B(A) = IA . Lemma 11. Let A be a FLw -algebra and let a, b ∈ IA with a ≤ b. Then the interval [a, b] = {x ∈ A : a ≤ x ≤ b} is a FLw -algebra, with 0 = a, 1 = b, ∧, ∨, · inherited from A, and x\y = (x\A y) ∧ b, x/y = (x/A y) ∧ b. If A is a GBL-algebra, then so is [a, b]. 8

Proof. As in Lemma 5, h(x) = xb is a homomorphism from A to Ab. For any integral residuated lattice B and idempotent a ∈ B, the principal filter ↑a is a subalgebra of B (see [8] Lemma 3.40). Therefore the GBL identity holds in [a, b] if it holds in A. We now generalize the poset decomposition result of [9] from finite GBLalgebras to FLw -algebras. Recall that an element W c in a lattice L is completely join irreducible if, for any subset S of L, c = S implies c ∈ S. Equivalently, c is completely join irreducible if there exists a unique element c∗ < c, called a lower cover of c, such that no element of L is strictly between c∗ and c. Theorem 12. Consider a FLw -algebra A with a finite subalgebra C such that C ⊆ IA , and let X be the dual of the partially ordered set of completely join Q irreducible elements of C. If Ac = ↓c∗ ⊕[c∗ , c] for all c ∈ X then A ∼ [c = X ∗ , c], where c∗ is the unique lower cover of c in C and [c∗ , c] is an interval in A. Proof. Let A be a FLw -algebra with a subalgebra QC that satisfies the assumptions of the theorem. We define the map h : A → X [c∗ , c] by h(a)(c) = ac ∨ c∗ (this is an element of [c∗ , c] since c∗ ≤ ac ∨ c∗ ≤ C). To see that f = h(a) is an element of the poset product, we have to show that if c < d in X (hence c > d in C) then f (c) = c∗ (the 0 of [c∗ , c]) or f (d) = d (the 1 of [d∗ , d]). Assuming f (c) 6= c∗ , we have a ∧ c > c∗ since Ac = ↓c∗ ⊕ [c∗ , c]. Therefore a > c∗ ≥ d, and it follows that f (d) = ad ∨ d∗ = d. We claim that h is a FLw -algebra isomorphism. It suffices to show that h is an order-isomorphism that preserves the monoid structure (since orderisomorphisms always preserve the first-order definable lattice operations and residuals). We have h(1) = 1 since 1c ∨ c∗ = c, and the preservation of · follows from (ac ∨ c∗ )(bc ∨ c∗ ) = acbc ∨ acc∗ ∨ bcc∗ ∨ c∗ = (ab)c ∨ c∗ . The map h is clearly order-preserving, and to show it is a bijection, we define Q g : X [c∗ , c] → A by _ g(f ) = {f (c) : f (k) = k for all k ∈ X with k